Tyrion makes it out of bed, finally. Bronn still seems loyal to him. He's now Ser Bronn. After getting up to date, Tyrion goes to moan at his father who does not seem sufficiently grateful to Tyrion for saving the city.
Tywin appears to know most of what Tyrion has done but isn't that impressed. When Tyrion asks to be Tywin's heir and inherit Casterly Rock (Jaime apparently can't since he's a kingsguard), Tywin says no. Tyrion might have low cunning but he sleeps with too many whores and anyway he's too ugly and deformed. It seems a bit harsh to begrudge him the whores, and anyway is it such a big deal with everything else the gentry of Westeros get up to?
